Let's analyze the patterns and how they evolve from stage to stage for both Ahmad and Chris.
### a. Ahmad's Design
Stage 1:
[tex]\[
\begin{array}{c}
[ \ ] \\
[█] \\
[ \ ]
\end{array}
\][/tex]
- Ahmad's design contains a single filled square in the center of the middle row.
Stage 2:
[tex]\[
\begin{array}{c c}
[ \ ] & [ \ ] \\
[█] & [ \ ] \\
[ \ ] & [ \ ]
\end{array}
\][/tex]
- Ahmad's design now has one filled square moved to the left and an empty square to the right.
Stage 3:
[tex]\[
\begin{array}{c c c}
[ \ ] & [ \ ] & [ \ ] \\
[█] & [█] & [█] \\
[ \ ] & [ \ ] & [ \ ]
\end{array}
\][/tex]
- In this stage, Ahmad’s design has expanded to three filled squares in a row.
Summary for Ahmad's Design:
- Stage 1: 1 filled square at the center
- Stage 2: 1 filled square to the left with an empty square to the right
- Stage 3: 3 filled squares in a row in the center
### b. Chris's Design
Stage 1:
[tex]\[
\begin{array}{c}
[█]
\end{array}
\][/tex]
- Chris's design starts with a single filled square.
Stage 2:
[tex]\[
\begin{array}{c c}
[█] & [█]
\end{array}
\][/tex]
- In Stage 2, Chris's design has expanded to two filled squares.
Stage 3:
[tex]\[
\begin{array}{c c c}
[█] & [█] & [█]
\end{array}
\][/tex]
- By Stage 3, Chris's design has expanded to three filled squares.
Summary for Chris's Design:
- Stage 1: 1 filled square
- Stage 2: 2 filled squares
- Stage 3: 3 filled squares
### Conclusion
To summarize:
- Ahmad's design changes from having a single central filled square in Stage 1, to having one filled square on the left with an empty square on the right in Stage 2, to three filled squares in the center row in Stage 3.
- Chris's design evolves more simply by just adding one filled square at each stage: starting with one square, then two, and finally three filled squares.
